Portfolio Optimization by Means of Meta-Resampled Efficient Frontiers

ABSTRACT

A computer-implemented method and computer program product for selecting a portfolio weight (subject to specified constraints) for each of a plurality of assets of an optimal portfolio. A mean-variance efficient frontier is calculated based on input data characterizing the defined expected return and the defined standard deviation of return of each of the plurality of assets. Multiple sets of optimization inputs are drawn from a distribution of simulated optimization inputs consistent with the defined expected return, the defined standard deviation of return of each of the plurality of assets and then a simulated mean-variance efficient frontier is computed for each set of optimization inputs. A meta-resampled efficient frontier is determined as a statistical mean of associated portfolios among the simulated mean-variance efficient frontiers, and a portfolio weight is selected for each asset from the meta-resampled efficient frontier according to a specified investment objective. The number of simulations and the number of simulation periods is determined on the basis of a specified level of forecast certainty.

The present application is a divisional application of copending U.S.patent application Ser. No. 11/158,267, issued as U.S. Pat. No.7,______, which was a continuation-in-part of U.S. Ser. No. 10/280,384,filed Oct. 25, 2002 and subsequently issued as U.S. Pat. No. 6,928,418.This application claims priority from the latter application. Bothpredecessor applications are incorporated herein by reference.

TECHNICAL FIELD

The present invention relates to methods for controlling thediscriminatory power of statistical tests of congruence between acurrent portfolio of tangible or intangible assets and a targetportfolio and of defining normal ranges of allocation to asset classeswithin a portfolio.

BACKGROUND ART

Managers of assets, such as portfolios of stocks, projects in a firm, orother assets, typically seek to maximize the expected or average returnon an overall investment of funds for a given level of risk as definedin terms of variance of return, either historically or as adjusted usingtechniques known to persons skilled in portfolio management.Alternatively, investment goals may be directed toward residual returnwith respect to a benchmark as a function of residual return variance.Consequently, the terms “return” and “variance,” as used in thisdescription and in any appended claims, may encompass, equally, theresidual components as understood in the art. The capital asset pricingmodel of Sharpe and Lintner and the arbitrage pricing theory of Ross areexamples of asset evaluation theories used in computing residual returnsin the field of equity pricing. Alternatively, the goal of a portfoliomanagement strategy may be cast as the minimization of risk for a givenlevel of expected return.

The risk assigned to a portfolio is typically expressed in terms of itsvariance σ_(p) ² stated in terms of the weighted variances of theindividual assets, as:

${\sigma_{p}^{2} = {\sum\limits_{i}{\sum\limits_{j}{{wiwj}\; \sigma \; {ij}}}}},$

where w_(i) is the relative weight of the i-th asset within theportfolio, σ_(ij)=σ_(i)σ_(j)ρ_(ij) is the covariance of the i-th andj-th assets, ρ_(ij) is their correlation, and σ_(i) is the standarddeviation of the i-th asset. The portfolio standard deviation is thesquare root of the variance of the portfolio.

Following the classical paradigm due to Markowitz, a portfolio may beoptimized, with the goal of deriving the peak average return for a givenlevel of risk and any specified set of constraints, in order to derive aso-called “mean-variance (MV) efficient” portfolio using knowntechniques of linear or quadratic programming as appropriate. Techniquesfor incorporating multiperiod investment horizons are also known in theart. As shown in FIG. 1A, the expected return μ for a portfolio may beplotted versus the portfolio standard deviation σ, with the locus of MVefficient portfolios as a function of portfolio standard deviationreferred to as the “MV efficient frontier,” and designated by thenumeral 10. Mathematical algorithms for deriving the MV efficientfrontier are known in the art.

Referring to FIG. 1B, a variation of classical Markowitz MV efficiencyoften used is benchmark optimization. In this case, the expectedresidual return α relative to a specified benchmark is considered as afunction of residual return variance ω, defined as was the portfoliostandard deviation σ but with respect to a residual risk. An investorwith portfolio A desires to optimize expected residual return at thesame level ω_(A) of residual risk. As before, an efficient frontier 10is defined as the locus of all portfolios having a maximum expectedresidual return α of each of all possible levels of portfolio residualrisk.

Known deficiencies of MV optimization as a practical tool for investmentmanagement include the instability and ambiguity of solutions. It isknown that MV optimization may give rise to solutions which are bothunstable with respect to small changes (within the uncertainties of theinput parameters) and often non-intuitive and thus of little investmentsense or value for investment purposes and with poor out-of-sampleaverage performance. These deficiencies are known to arise due to thepropensity of MV optimization as “estimation-error maximizers,” asdiscussed in R. Michaud, “The Markowitz Optimization Enigma: IsOptimized Optimal?” Financial Analysts Journal (1989), which is hereinincorporated by reference. In particular, MV optimization tends tooverweight those assets having large statistical estimation errorsassociated with large estimated returns, small variances, and negativecorrelations, often resulting in poor ex-post performance.

Resampling of a plurality of simulations of input data statisticallyconsistent with an expected return and expected standard deviation ofreturn has been applied (see, for example, Broadie, “Computing efficientfrontiers using estimated parameters”, 45 Annals of Operations Research21-58 (1993)) in efforts to overcome some of the statisticaldeficiencies inherent in use of sample moments alone. Comprehensivetechniques based on a resampled efficient frontier are described in U.S.Pat. No. 6,003,018 (Michaud et al. '018), issued Dec. 14, 1999, and inthe book, R. Michaud, Efficient Asset Management, (Harvard BusinessSchool Press, 1998, hereinafter “Michaud 1998”), that MV optimization isa statistical procedure, based on estimated returns subject to astatistical variance, and that, consequently, the MV efficient frontier,as defined above, is itself characterized by a variance. The Michaudpatent and book are incorporated herein by reference, as are allreferences cited in the text of the book.

As taught in the Michaud '018 patent, an MV efficient frontier is firstcalculated by using standard techniques as discussed above. Since theinput data are of a statistical nature (i.e., characterized by meanswith associated variances and other statistical measures), the inputdata may be resampled, by simulation of optimization input parameters ina manner statistically consistent with the first set of data, asdescribed, for example, by J. Jobson and B. Korkie, “Estimation forMarkowitz Efficient Portfolios,” Journal of Portfolio Management,(1981), which is herein incorporated by reference. Embodiments of thepresent invention are related to improvements and extensions of theteaching of the Michaud '018 patent.

When portfolios are rebalanced in accordance with current practice,criteria are applied that are typically not portfolio-based orconsistent with principles of modern statistics but are generallyassociated with various ad hoc rules. U.S. Pat. No. 6,003,018 teaches aportfolio-based rebalancing criterion that can be used for allportfolios on the resampled efficient frontier and that is consistentwith principles of modern statistics and considers the uncertainty ininvestment information.

Michaud 1998, provided data, for purposes of illustration, thatconsisted of 18 years of monthly returns for eight asset classes. Theresampling process illustrated in the text computes simulated efficientfrontiers of 18 years of returns, or 216 monthly resampled returns foreach set of simulated means and covariances and associated simulatedefficient frontiers, prior to obtaining the average. In this instancethe resampling of returns duplicates the amount of information in thehistorical return dataset. It is desirable to allow for other variableassessments of confidence in the forecasting power of the data, and thatis addressed, below, in the context of the present invention.

Other features of rebalancing procedures, as practiced heretofore,imposed important limitations. The discriminatory power was notcustomizable, with too high power at low levels of risk and too littlepower at high levels of risk. Methods are clearly necessary forproviding relatively uniform discriminatory power across portfolio risklevels as well as being able to customize discriminatory power accordingto the investment needs of organizations which differ in terms of usersophistication, asset class characteristics, or investment strategyrequirements. Methods are also clearly desirable that help identifyanomalously weighted assets (overly large or small weights) relative toa normal range that is associated with the uncertainty of investmentinformation.

SUMMARY OF THE INVENTION

In accordance with preferred embodiments of the present invention, acomputer-implemented method is provided for selecting a value of aportfolio weight for each of a plurality of assets of an optimalportfolio. The value of portfolio weight is chosen from specified valuesassociated with each asset, between real numbers c₁ and c₂ that may varyby asset, each asset having a defined expected return and a definedstandard deviation of return. Each asset also has a covariance withrespect to each of every other asset of the plurality of assets. Themethod has steps of:

a. computing a mean-variance efficient frontier based at least on inputdata characterizing the defined expected return and the defined standarddeviation of return of each of the plurality of assets;

b. indexing a set of portfolios located on the mean-variance efficientfrontier thereby creating an indexed set of portfolios;

c. choosing a forecast certainty level for defining a resampling processof the input data consistent with an assumed forecast certainty of theinput data;

d. resampling, in accordance with the process defined by the forecastcertainty level, a plurality of simulations of input data statisticallyconsistent with the defined expected return and the defined standarddeviation of return of each of the plurality of assets;

e. computing a simulated mean-variance efficient portfolio for each ofthe plurality of simulations of input data;

f. associating each simulated mean-variance efficient portfolio with aspecified portfolio of the indexed set of portfolios for creating a setof identical-index-associated mean-variance efficient portfolios;

g. establishing a statistical mean for each set ofidentical-index-associated mean-variance efficient portfolios, therebygenerating a plurality of statistical means, the plurality ofstatistical means defining a resampled efficient frontier,

wherein processes (a), (b), and (d)-(g) are digital computer processes;

h. selecting a portfolio weight for each asset from the resampledefficient frontier according to a specified utility objective; and

i. investing funds in accordance with the selected portfolio weights.

In accordance with other embodiments of the invention, the specifiedutility objective may be a risk objective, and steps (d) through (i) maybe repeated based upon a change in a choice of forecast certainty level.The forecast certainty level may be based upon investment horizon orother factors.

In yet other embodiments of the invention, the step of indexing the setof portfolios may include associating a rank with each portfolio of theindexed set of portfolios located on the mean-variance efficientfrontier, and may also include associating a parameter for which aspecified measure of expected utility is maximized, one possiblespecific measure of expected utility represented by a quantity μ−λσ²,where σ² is the variance of each portfolio and μ is the defined expectedreturn of each portfolio of the set of portfolios located on themean-variance efficient frontier.

In accordance with another aspect of the present invention,computer-implemented methods are provided for selecting a value of aportfolio weight for each of a plurality of assets of an optimalportfolio. In these embodiments, the methods have steps of:

-   -   a. computing a mean-variance efficient frontier, the frontier        having at least one mean-variance efficient portfolio, based at        least on input data characterizing the defined expected return        and the defined standard deviation of return of each of the        plurality of assets;    -   b. choosing a forecast certainty level from a collection of        forecast certainty levels for defining a resampling process of        the input data consistent with the assumed forecast certainty of        the input data;    -   c. generating a plurality of optimization inputs drawn at least        from a distribution of simulated optimization inputs consistent        with the defined expected return, the defined standard deviation        of return of each of the plurality of assets and the chosen        forecast certainty level;    -   d. computing a simulated mean-variance efficient frontier, the        frontier having at least one simulated mean-variance efficient        portfolio, for each of the plurality of optimization inputs;    -   e. associating each mean-variance efficient portfolio with a        specified set of simulated mean-variance efficient portfolios        for creating a set of associated mean-variance efficient        portfolios;    -   f. establishing a statistical mean for each set of associated        mean-variance efficient portfolios, thereby generating a        plurality of statistical means, the plurality of statistical        means defining a meta-resampled efficient frontier;    -   g. selecting a portfolio weight for each asset from the        meta-resampled efficient frontier according to a specified        investment objective; and    -   h. investing funds in accordance with the specified portfolio        weights.

In various alternate embodiments of the invention, the specifiedinvestment objective may be a risk objective. The step of choosing aforecast certainty level from a collection of forecast certainty levelsmay include choosing from a set of indices from 1 to N, where each indexis calibrated to provide a different level of forecast certainty. Thenumber of samples drawn from a distribution may be increased based on ahigher level of forecast certainty. Each level of forecast certainty mayrepresent a geometric increase (or decrease) in the number ofobservations drawn from the distribution.

In further alternate embodiments of the invention, the step ofassociating each mean-variance efficient portfolio with a specified setof simulated mean-variance efficient portfolios may include associatingon the basis of proximity to a maximized expected utility, where, for aspecific example, the expected utility may be a quantity μ−λσ².

The step of associating each mean-variance efficient portfolio with aspecified set of simulated mean-variance efficient portfolios mayinclude associating on the basis of a measure of risk, where the measureof risk may be variance or residual risk.

In accordance with yet further embodiments of the invention, the step ofassociating each mean-variance efficient portfolio with a specified setof simulated mean-variance efficient portfolios may include associatingon the basis of a measure of return such as expected return or expectedresidual return. The step of associating each mean-variance efficientportfolio with a specified set of simulated mean-variance efficientportfolios may also include associating on a basis of proximity of theportfolios to other simulated mean-variance efficient portfolios orassociating on a basis of ranking by a specified criterion, thecriterion chosen from a group including risk and expected return.Alternatively, the step of associating each mean-variance efficientportfolio with a specified set of simulated mean-variance efficientportfolios may include associating on a basis of investment relevance ofthe simulated mean-variance efficient portfolios or associatingsimulated portfolios based on a selection of relevant simulations.

The step of associating each mean-variance efficient portfolio with aspecified set of simulated mean-variance efficient portfolios mayinclude specifying a fraction of portfolios per simulation to beconsidered in the association.

Additional steps of the method may include calculating an estimatednormal range of asset weights for an optimal portfolio and calculating aportfolio rebalancing probability for a given portfolio with respect toan optimal portfolio.

In yet another aspect of the present invention, a computer programproduct is provided for use on a computer system for selecting a valueof portfolio weight for each of a specified plurality of assets of anoptimal portfolio. The computer program product has a computer usablemedium having computer readable program code thereon, with the computerreadable program code including:

-   -   a. program code for computing a mean-variance efficient frontier        based at least on input data characterizing the defined expected        return and the defined standard deviation of return of each of        the plurality of assets;    -   b. program code for aggregating a set of portfolios located on        the mean variance efficient frontier;    -   c. a routine for generating a plurality of optimization inputs        drawn from a distribution of simulated optimization inputs        statistically consistent with the defined expected return and        the defined standard deviation of return of each of the        plurality of assets;    -   d. program code for computing a simulated mean-variance        efficient frontier, the frontier comprising at least one        simulated mean-variance efficient portfolio for each of the        plurality of optimization inputs;    -   e. program code for associating each simulated mean-variance        efficient portfolio with a specified portfolio of the set of        aggregated portfolios for creating a set of associated        mean-variance efficient portfolios;    -   f. a module for establishing a statistical mean for each set of        associated mean-variance efficient portfolios, the plurality of        statistical means defining a meta-resampled efficient frontier;        and    -   g. program code for selecting a portfolio weight for each asset        from the meta-resampled efficient frontier according to a        specified risk objective.        In alternative embodiments, the routine for resampling a        plurality of simulations on input data may draw a number of        returns from a simulation based on a specified level of        estimation certainty, and the set of associated mean-variance        efficient portfolios may be based on a specified level of        forecast certainty.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be more readily understood by reference to thefollowing description, taken with the accompanying drawings, in which:

FIG. 1A depicts the prior art principle of calculating an efficientfrontier of maximum expected return for each given level of portfoliorisk;

FIG. 1B depicts the prior art principle of calculating an efficientfrontier of maximum expected residual return for each given level ofportfolio residual risk;

FIG. 2 displays a set of statistically equivalent portfolios within therisk/return plane;

FIGS. 3A-3F represent constituent modules of a flow-chart depicting aprocess for computing a portfolio rebalancing index, in accordance withembodiments of the present invention;

FIG. 4 displays statistically equivalent portfolios within therisk/return plane corresponding to three particular risk rankings on theefficient frontier: minimum variance, maximum return, and a middlereturn portfolio;

FIG. 5 shows the resampled efficient frontier plotted in the risk/returnplane in accordance with a preferred embodiment of the presentinvention;

FIG. 6 shows the classical and several resampled efficient frontiers,illustrating various levels of estimation certainty in accordance withembodiments of the present invention;

FIG. 7 is a plot showing an identified portfolio on a resampledefficient frontier based on a specified maximum expected utility point;

FIG. 8A is a plot showing a typical need-to-trade probability for aspecified current portfolio relative to a specified target portfolio asa function of a specified portfolio risk;

FIG. 8B is a plot showing need-to-trade probabilities for a threespecified current portfolios relative to corresponding specified targetportfolios as a function of a specified portfolio risk, using classicalefficient portfolios; and

FIG. 8C is a plot showing need-to-trade probabilities for a threespecified current portfolios relative to corresponding specified targetportfolios as a function of a specified portfolio risk, using resampledefficient portfolios in accordance with embodiments of the presentinvention.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

MV-efficient portfolios may be recalculated, based on distinctresamplings of data, subject to the same constraints as applied in theinitial solution. Inputs to each solution include the forecast returnsand standard deviations and distributional assumptions of returns drawn,typically, for a plurality of asset classes (stocks, mutual funds,currencies, etc.) from a multivariate distribution. Correlations acrossasset classes are also input.

In accordance with preferred methods for resampling when no additionaldistributional information is known, simulated returns are drawn for aplurality of asset classes from a multivariate normal distribution.Other resampling methodologies may entail bootstrapping of historicaldata or resampling under distributional assumptions (for instance, ifskew or kurtosis of returns is known). This Monte Carlo simulationgenerates different forecasts of risk and return, including, but notlimited to, mean, standard deviation, and correlation, to be used ineach computation of MV-efficient portfolios.

As an alternative procedure when the distribution of simulatedoptimization inputs is known, resampled optimization input parameters(mean return, standard deviation, correlations, etc.) may be drawn fromtheir distribution. For example, if multivariate normality of returns isassumed and the vector of returns r for all assets is modeled byr˜N(μ,Σ), and a forecast certainty corresponding to N observations ofreturn is given, in each simulation i, a forecast mean return μ_(i) andcovariance of returns Σ_(i) can be modeled by:

μ_(i)˜N(μ,Σ/N)

Σ_(i)˜Inv-Wishart_(N-1)(Σ),

where Wishart distributions, their notation, and their application tounknown covariance matrices are discussed in detail in Gelman et al.,Bayesian Data Analysis, Chapman & Hall (1995), which is incorporatedherein by reference.

The constraints applied to each of the solutions, given the aforesaidoptimization inputs, include constraints placed on the set of weightsaccorded to the various components of the portfolio. An efficientsolution may be sought in which the solution is a set of weights{w_(i)}of assets comprising a portfolio or, else, a set of activeweights {x_(i)}of assets defined differentially with respect to abenchmark portfolio. In the latter case, the benchmark portfolio isdesignated, in FIG. 1B, as point ‘A’ with respect to which residual riskand expected residual return are plotted in the figure. In either case(i.e., whether the weights are to be solved for absolutely or withrespect to a benchmark portfolio), the weights are subject toconstraints specified by the user. These constraints may include, forexample, a constraint that one or more specified weights, or the sum ofspecific weights, must lie between two specified real numbers, c₁ andc₂. For example, c₁ and c₂ may correspond, respectively, to 0 and 1. Theinclusion of negative weights allows the resampled frontier, discussedbelow, to include portfolios of both long and short asset weights.Similarly, the sum of portfolio weights may be constrained, for example,by requiring the sum to equal an amount to be invested, or,conventionally, 1.

Based on multiple resamplings, as shown in FIG. 2, a set 12 ofstatistically equivalent MV efficient portfolios may be calculated. Byiterating this procedure, a large MV efficient “statistical equivalence”set of portfolios, in the expected return—portfolio variance space, maybe generated. Multiple resamplings may be based upon returns drawn aspecified number of times from an assumed distribution for a particularasset class. All a priori sets of assumptions with respect to thedistribution are within the scope of the present invention; for example,the distributions may be defined by bootstrapping, normal, log-normal,mixed, etc. As an alternative algorithm, statistical input parameters(mean return, standard deviation, correlations) may be derived from aset of returns drawn from a particular simulation, and those parameters,in turn, used, in a bootstrapping manner, for subsequent resamplings,thus resampled distributions may serve the same function as resampledreturns with respect to derivation of a resampled efficient frontierthat may be referred to as a ‘meta-resampled efficientfrontier.’Referring now to FIGS. 3A-3F, a process is described forderiving a Portfolio Rebalancing Index, which describes the percentilelevel of a portfolio norm relative to a distribution of portfolio norms,as will be described below.

As shown in FIG. 3A, a set of initial Optimization Inputs, based onhistorical performance data relative to a set of assets, or otherwise,is provided by a user. Based on any such Optimization Inputs, a set of Kmean-variance (MV) optimized portfolios may be calculated usingclassical Markowitz optimization. These K mean-variance optimizedportfolios span the mean-variance efficient frontier for a given set ofoptimization inputs, in that they encompass the range either fromminimum low to maximum high risk or minimum low to maximum high expectedreturn. Any one of these portfolios may be referred to herein, and inany appended claims, as optimal portfolios, understanding that theoptimization criterion might be based upon any specified utilityfunction and not just one having a preference for mean variance alone.Optimization criteria may be based on specified investment objectivessuch as portfolio risk or a specified utility function. The set of Kportfolios spanning the mean-variance efficient frontier for a given setof optimization inputs will be referred to as ‘K-MV’.

At this stage, three parameters may be defined, which will have therespective significance as now indicated:

-   -   N designates the total number of resampled sets of optimization        inputs of the initial optimization inputs;    -   M designates the number of resampled sets of optimization inputs        of a given set of resampled optimization inputs; and

M2 designates the number of randomly chosen K-MV portfolios from the Nsimulated K-MV portfolios.

The running index i is the index over which successive variables arestepped for successive sets of resampled optimization inputs in derivinga distribution of portfolio norms.

Until the process is complete, i.e., as long as i is less than N, as iis incremented, a set of resampled inputs is created in the step labeledResample Inputs. In this step, a new set of risk and return inputs iscreated for a given level of confidence in the inputs. The distributionfor the resampled inputs is based on the original inputs (or theprevious resampled inputs when resampling inputs in module B, asdescribed below). An efficient frontier (designated EFF(i)) is derivedfor each new set of resampled inputs, namely the set K-MV describedabove, and the same set is also stored as a set of Equivalent PortfoliosEQ(i).

Module B (FIG. 3B) performs N successive resamplings and stores the K-MVefficient frontier portfolios in the array EFF. Once this has beencompleted, Module C (FIG. 3C) first associates the portfolios of thesimulated K-MV with portfolios from other K-MV efficient portfolios.Various procedures for associating portfolios of K-MV sets are discussedbelow, whether by rank order value from each K-MV, or otherwise. Onceportfolios from the various K-MVs have been associated, they areaveraged (again, in a generalized sense, as described below) in order toderive the resampled efficient frontier (designated REF).

As shown in FIG. 3D, if M2 is non-zero, associated portfolios arebootstrapped, that is to say, M2 associated portfolios are randomlyselected from the set of simulated equivalent portfolios and averaged toform a new set of equivalent portfolios EQ(i).

Once the new set of equivalent portfolios is set up, statistics may beperformed on the asset weights of the K portfolios of the ResampledEquivalent Frontier with respect to the distribution of the associatedasset weights, including their respective confidence percentiles, andother statistical measures, as shown in FIG. 3D.

Referring, finally, to FIG. 3F, portfolio norms are computed, asdiscussed further below, for each simulated portfolio with respect toevery portfolio (or a relevant subset) of the resampled efficientfrontier. The various norms that may be employed are discussed below,but one norm entails the tracking error between two specifiedportfolios. In order to gauge the advisability of a portfoliorebalancing, for all K portfolios of the resampled efficient frontier, aset of simulated portfolios is chosen from EQ(i) that represents thosedeemed to be the most reasonable alternative investments. In a preferredembodiment of the invention, some fraction, typically about 5%, of theportfolios are selected, where this fraction encompasses thosesimulations with the smallest portfolio norm with respect to eachcorresponding portfolio of the resampled efficient frontier. Theportfolio rebalancing index may then be determined as the percentilelevel of portfolio norm for a specified portfolio with respect to theportfolio norm distribution.

Rather than resampling based on historical returns, a furtheralternative procedure entails factor model resampling, wherein the setof returns is modeled according to a linear (in this example) model,such as:

r _(i)=α_(i)+β_(i) ′F+(any other mod eled terms)+ε_(i),

where, for each asset ‘i’, α signifies asset-specific excess return, Fsignifies the vector of factor returns for all assets (be they returnsassociated with a representative portfolio, stylized factor, etc.),β′_(i) signifies a (transposed) vector of coefficients to any factorreturn for asset ‘i’ (be they estimated historical, or, alternatively,current characteristic values of the asset, e.g., dividend-to-priceratio), and is a residual stochastic component of the return specific toasset ‘i’. Factor returns may be modeled in a variety of ways, buttypically each represents pervasive market factors, and, in this model,represents any variable influencing return of two or more assets.Statistical inputs may be specified for distributions of each of theconstituent terms, whether based on systematic or idiosyncraticvariance, which constituent terms may accordingly then be resampled.

Based on application of a specified statistical procedure, an existingportfolio (or, the “CURRENT portfolio”) may be found to be consistent,in a statistical sense, with efficiency and thus not requireoptimization, thereby potentially and advantageously saving transactioncosts associated with revision of a portfolio. Alternatively, based onthe same statistical procedure, an existing portfolio may be found to beinconsistent with efficiency and an alert produced indicating a need torebalance to the appropriate target optimal portfolio. Implementation ofa rebalancing test is described in detail below. Referring to FIG. 4,statistically equivalent portfolios within the risk/return plane areshown corresponding to three particular risk rankings on the efficientfrontier: namely, minimum variance 14, maximum return 16, and a middlereturn portfolios 18.

In the context of statistical equivalence of portfolios, a“when-to-trade probability” or “rebalance probability” is defined as theconfidence level of rejection test for portfolio statisticalequivalence. More particularly, with respect to an “optimal resampledefficient portfolio,” described in detail below, the “when-to-tradeprobability” is defined as the percentage of resampled portfolios“closer” (in terms of a norm to be discussed below) to the “optimalresampled efficient portfolio” than the portfolio in question, i.e., theCURRENT portfolio. The norm typically employed is that of the varianceof a portfolio with respect to a particular optimal resampled efficientportfolio (commonly referred to as tracking error), however other normsare within the scope of the present invention.

Referring further to FIG. 4, all resampled portfolios within therisk/return plane may be associated, many-to-one, with particularportfolios on MV efficient frontier 10. Various criteria may be appliedin associating portfolios with those on the MV efficient frontier, andall such associations are within the scope of the present invention. Asone example, each of the K efficient frontier portfolios (i.e., eachpoint on efficient frontier 10) may be identified by its relative returnrank. Similarly, the efficient frontier portfolios may be ranked bytheir variance, the maximum variance corresponding to the maximumreturn, the rankings by risk or return similarly mapping onto oneanother uniquely. Thus, for example, the minimum variance portfolio 14might have the lowest rank relative to the other efficient portfolios ofefficiency frontier 10. Similarly, maximum average return portfolio 16has the highest average return rank in each simulated efficientfrontier. Similarly, any other simulated portfolio is rank associatedwith a particular efficient frontier portfolio. The sparsely clusteredportfolios 18 shown in the figure correspond to the ‘middle’ rankedefficient portfolio. In practice, the shape of the rank-associatedregions varies in dependence upon the position of the portfolio on theMV efficient frontier.

It is not necessary, however, that the association with efficientfrontier portfolios be by rank, and particular portfolios on the MVefficient frontier may be indexed, and thus index-associated, each witha set of statistically equivalent efficient portfolios lying below theefficient frontier. Indexing, for example, of the set of MV efficientportfolios may be by associating with each MV efficient portfolio a“lambda value,” defining the risk/return preference, with respect towhich the quantity φ=σ²−λμ is minimized, where σ² is the variance ofeach portfolio and μ is the expected return of each portfolio of the setof portfolios located on the mean variance efficient frontier. Theparameter λ assumes a value between zero and infinity. The foregoingprocedure is mathematically equivalent to maximizing the quantity μ−λσ².

Moreover, association of efficient portfolios for deriving an average(in a specified sense), and thus a resampled efficient frontier, are notrequired to be index-ranked at all, within the scope of the presentinvention. Indeed, in alternate embodiments of the invention,‘neighboring’, or otherwise related, portfolios may be grouped, in orderto achieve desired aggregation of portfolio characteristics.

Methods for association for averaging portfolios relative to an indexset include equal or weighted averages by significance or other nearnessmeasures where the index set defined for all simulated portfolios in theresampling process. An example of an index set may include the set ofsimulated portfolios with similar utility values. In a further example,a resampled frontier may represent a best-fit curve through portfoliospace of simulated portfolios with weighted averages taken over somespecified subset.

An alternative means of associating portfolios from distinct ensemblesof portfolios, say, for example, as derived from successive resamplings,may be on the basis of maximizing expected utility. Referring to FIG. 7,an efficient frontier 60 is shown in the risk-return plane. Variousutility functions 62, 64, and 65 are plotted as may be specified forparticular investors. Each plotted utility function curve, say 64 forexample, represents a constant utility for the specified investor.Utility function 62, for example, obeys a different functional law fromthat of utility function 64. Functional forms typically employed may beexponential and/or polynomial functions of risk.

Utility functions 64 and 65 obey an identical functional dependence ofreturn vs. acceptable risk, differing only in the total utility,indicated, by way of example, by the quantity u which assumes the value5 and 6 for curves 64 and 65 respectively. The point 68 on efficientfrontier 60 may be characterized in that it represents a portfolio thatmaximizes the expected utility with respect to the class of utilityfunctions to which curves 64 and 65 belong. Similarly, point 66 onefficient frontier 60 may be characterized in that it represents aportfolio that maximizes the expected utility with respect to the classof utility functions to which curve 62 belongs. Other points onefficient frontier 60 may be characterized in terms of their ‘distance’(as defined by a specified norm) from point 68 of maximum expectedutility. Similarly, points on different efficient frontiers may beassociated, as described herein in the context of defining a resampledefficient frontier, on the basis of their identity or proximity toportfolios maximizing expected utility with respect to specified utilityfunctions.

Rebalancing of a portfolio is indicated if the current portfolio isstatistically distinct from a target optimized portfolio on a resampledefficient frontier identified according to criteria to be discussedbelow. “Proximity” of one portfolio (whether resampled, indexed orotherwise) to a corresponding portfolio may be defined in terms of atest metric based on a “norm,” with the norm having the usual propertiesof a distance function as known to persons skilled in the mathematicalarts. The properties of a norm defined in a vector space are well-known:a norm associates a non-negative value with any vector in the space,preserves scalar multiplication, and obeys the relation ∥x+y∥≦∥x∥+∥y∥.

Various norms may be used for defining distance in the risk/returnspace. The distance criterion for any portfolio P is typically taken tobe the relative variance for portfolio P,

(P−P₀)^(t)*S*(P−P₀),

where P−P₀ is the difference vector of portfolio weights with respect toP₀, the corresponding index-associated portfolio on the resampledefficient frontier, and, S is the input return covariance matrix (withthe superscript ‘t’ denoting the transpose of the difference vector).The norm is taken in the space of portfolio vectors (i.e., “portfoliospace”). Alternative distance criteria may include additional functions(linear or otherwise) of P−P₀.

The CURRENT portfolio is measured for statistical distinction against aset of simulated portfolios, the enhanced discriminatory ‘power’ of thebalancing test reflecting the decreasing likelihood that thedesirability of rebalancing a portfolio is ‘missed.’ I.e., a morepowerful rebalancing test is less likely to attribute a CURRENTportfolio to a distribution of like portfolios where it properly doesnot belong to the population. Alternatively, controlling discriminatorypower enhances the likelihood of rebalancing only where appropriate.

Referring to FIG. 8A, an example is shown in which the need-to-tradeprobability for a given CURRENT portfolio, as determined by a particularrebalancing test, is plotted as a function of portfolio risk. The plotof FIG. 8A thus reflects the enhanced discriminatory power of thespecified rebalancing test. The CURRENT portfolio may have an associatedresidual risk close to a specified value σ_(A), thereby lyingsufficiently close to a specified efficient frontier as not to bestatistically distinguishable from a target portfolio. The need-to-tradeprobability may be close to zero over some range, whereas, outside thatrange, the probability of a given test dictating a need to rebalancerises in case it is desirable to achieve a different level of risk.

The portfolios against which a CURRENT portfolio is tested are typicallyweighted in performing a statistical rebalancing test. In accordancewith embodiments of the present invention, a set of portfolios isretained for performing the statistical test of the CURRENT portfoliowith the objective of increasing the discriminatory power of thestatistical test. An advantageous benefit that this procedure mayprovide is that of substantial uniformity of power of the rebalancingtest across the entire efficient frontier. This may advantageously beachieved by weighting sample portfolios increasingly with proximity to atarget portfolio. Additionally, this procedure may be usedadvantageously to reduce computational overhead in performing arebalancing test.

Once an ensemble of MV efficient portfolios has been associated, whetherby index-set association, proximity to a maximum expected utility point,or other method of aggregation, usual statistical measures of theensemble may be derived. These measures include, without limitation, theaverages, standard errors, and t-statistics of the average of theportfolio weights of the rank-associated simulated efficient portfolios.Referring now to FIG. 5, an average of index-associated MV efficientportfolios may be defined, in accordance with preferred embodiments ofthe present invention, the average of index-associated MV efficientportfolios being referred to as a “resampled-efficient portfolio.” Theaverage may be determined with respect to any of a variety ofparameters, and, in accordance with a preferred embodiment, it is withrespect to the vector average of the associated portfolios. The vectoraverage of a set of portfolios is defined as the average over theweighted assets of each of the portfolios of the set, taking intoaccount the sign, positive or negative, of the contribution of aparticular asset to a particular portfolio. The locus 40 ofresampled-efficient portfolios is referred to as the “resampledefficient frontier.” The resampled-efficient portfolio and itsassociated statistics may be applied as a statistical measure forportfolio analysis, as further described herein. Its application, as achoice for portfolio selection, advantageously removes, by definition,the “outlier” portfolios which strongly depend on values of a particularset of inputs and improves out-of-sample performance, on average.Statistical procedures for portfolio analysis and revision, andperformance benefits based on these concepts, are further described indetail in the Michaud book.

Forecast Certainty Levels

In particular, the number of resampling simulations that are performedand efficient frontiers that are computed may be specified, prior toobtaining the average, or resampled efficient frontier, as discussed inthe foregoing paragraph. The number of resamplings of returns performedto compute simulated means and covariances and associated simulated MVefficient frontiers prior to obtaining the average, or resampledefficient frontier, is a free parameter of the resampled efficiencyoptimization procedure.

As discussed in the Background Section, above, Michaud 1998 taughtresampling of returns that duplicated the amount of information orforecast certainty in the historical return dataset. In the same way,for any historical return data set of assets, the implicit informationor forecast certainty of the dataset can be replicated by setting thenumber of resamplings or bootstraps of returns to equal the number ofhistorical returns in the data set.

In practice, the implicit information or forecast certainty of ahistorical return dataset may not be appropriate. For example, theinformation level of a 100 year monthly historical return dataset may beinconsistent with computing an optimal portfolio for the followingmonth. Alternatively, the means and covariances used to compute anoptimal portfolio is often not based on historical return data. In thatcase the number of resamplings of returns to compute simulated efficientfrontiers prior to obtaining the average or resampled efficient frontiermust be found from other considerations. In addition, the informationlevel associated with the means and covariances used to compute anoptimal portfolio or efficient frontier will often vary depending ontime period, outlook, and many other considerations even when the inputsare unchanged.

The number of resamplings of returns that are performed for computingsimulated means and covariances and associated efficient frontiers is anatural framework for modeling forecast certainty in the optimizationprocess. As the number of resamplings increases without limit, thesimulated means and covariances and associated efficient frontierapproaches the classical MV efficient frontier and the average orresampled efficient frontier approaches the classical Markowitz MVefficient frontier. The limit is the case of complete certainty in theoptimization inputs and the classical MV efficient frontier.

Referring now to FIG. 6, higher numbers of returns drawn from asimulation correspond to higher levels of estimation certainty. In FIG.6, the uppermost efficient frontier 10 is, as before, the classicalefficient frontier, determined on the basis of the original inputsassociated with each asset. Resampled efficient frontiers 50, 52, 54, 56and 58, correspond to increasing numbers of simulations (or numbers ofsimulations periods per simulation), and thus to increasing levels ofestimation certainty.

Conversely, as the number of resamplings of returns decreases, thesimulated means and covariances and efficient frontiers will vary widelyand the average or resampled efficient frontier will increasinglyresemble the no-information equal weighted or benchmark weightedportfolio. The limit is the case of complete uncertainty and equal orbenchmark portfolio weighting.

More particularly, in accordance with embodiments of the presentinvention, forecast certainty may also vary among input data sets, withparticular data sets giving rise to greater forecast certainty inpredicted performance than other input data sets. Optimization inputsgenerally reflect a forecast or expected return process assumed to have,on average, a positive correlation with ex post returns. The level ofinformation is typically expressed as the “information correlation”(IC)—e.g., an IC of 0.2 reflects an expected correlation of 0.2 offorecast with ex post returns.

Information correlation refers to the assumed correlation between aforecast and ex post actual return. IC typically varies with firmstrategy, industry, sector, etc. and may be stated, generally, to be aproxy for relative forecast certainty associated with a particular dataset. More generally, forecast certainty may be associated with the levelof information correlation (IC) and also by the standard deviation ofthe IC distribution

Forecast certainty level is defined with respect to the number ofresampling returns used to compute the simulated means and covariancesand associated efficient frontiers in the resampled efficient frontierprocess. Forecast certainty level shows that the resampled efficientfrontier is a generalization of classical mean-variance efficiency thatallows the manager to control the amount of confidence in the inputs inthe optimization process. At one extreme of confidence, resampledefficiency optimization is Markowitz efficiency; at the other extreme ofconfidence resampled efficiency results in either equal weighting or, inthe presence a benchmark index, the benchmark portfolio.

One purpose of the resampling procedure is to include an appropriatelevel of forecast uncertainty into the resampled optimization inputs. Tothat end, resampled inputs may be designed to reflect a range offorecast certainty levels. The resampling parameter N provides a meansof controlling the level of certainty implicit in the forecast process.

In accordance with preferred embodiments of the present invention, anindex value, assuming a numerical value between 1 and N, is associatedwith a forecast certainty attributed by an analyst to a data set. Theindex value corresponds to the number of resamplings for a particularsimulation. Thus, the more resamplings, the higher the forecastcertainty, with higher index values corresponding to successiveresampled efficient frontiers, 50, 52, 54, 56 and 58, as depicted inFIG. 6.

In a preferred embodiment, ten levels of certainty may be defined, where1 represents very uncertain and ten represents high but not perfectforecast certainty. In this preferred embodiment, an “average” level offorecast certainty is chosen for a particular application. For example,for equity portfolio optimization, level 4 may be defined ascorresponding to a level of certainty consistent with the way manycommercial risk models are estimated. In this case level 4 may bedefined as drawing five years or sixty months of monthly observationsfrom the monthly distribution. Other certainty levels associated witheach level represent a geometric increase (or decrease) in the number ofobservations drawn from the distribution (e.g. if level 4 corresponds to60 monthly observations, level 5 may correspond to 90 monthlyobservations and level 6 to 135 monthly observations).

The forecast certainty level assumption impacts all the resampledefficient frontier computations. Forecast certainty level defines theresampled efficient frontier and optimized portfolios, the resampledneed-to-trade probability and asset weight ranges. Forecast certaintylevel may vary over time to dynamically control the optimized portfolioor resampled efficient frontier, need-to-trade probability, and assetrange monitoring rules.

The way a manager chooses a level of forecast certainty is subjectiveand context dependent. The level may be associated with the time horizonof the optimization. For example, an optimal portfolio that isappropriate for a long-term investment strategy is likely to have adifferent forecast certainty level than one for short-term activemanagement. Intuitively, it is more likely that stocks will beat bondsover a ten year horizon than for any given month. Different strategies,information sets, and different time periods will drive the choice offorecast certainty level. But some choice of forecast certainty leveland resampled efficiency optimization is almost always advisable. Thisis because no investor, in practice, is ever 100% certain of theirinformation. MV optimization does not include any sense of estimationerror and essentially requires a metaphysical forecast certainty levelfor application.

Refined Discrimination Power of Rebalancing Tests

It is a characteristic of classical efficient frontiers that portfoliosat the high-risk extreme tend to be concentrated—specifically, with highweighting of assets bearing high expected returns. On the other hand,portfolios of resampled efficient frontiers at the correspondinghigh-risk extreme tend to be substantially diversified.

FIG. 8B shows that this leads to unsatisfactory statistical implicationsif a target portfolio based on resampled optimizations is compared withportfolios on a classical efficient frontier. In particular, for thecase of low-risk portfolios, points plotted along curve 72 show aneed-to-trade (i.e., a probability near unity) in almost all instances,where classical efficient portfolios are employed, since optimizationwill emphasize a particular low-risk target asset. Similarly, for thecase of high-risk portfolios, classical efficient portfolios will alsoindicate a need-to-trade in most cases, as shown along curve 76, in thiscase because maximum-return assets will be emphasized. A convenientmethod for calibrating the portfolio rebalancing procedure is to selecta set of portfolios that are considered to be statistically equivalentto an optimal portfolio and adjust the free parameters so as to includethe set of portfolios in the statistically equivalent region for anappropriate acceptance level. Curve 74 depicts the case of medium-riskportfolios indicating little discriminatory power with respect to highrisk efficient portfolios.

FIG. 8C shows that need-to-trade probabilities are ameliorated at boththe low-risk (curve 82) and high-risk (curve 86) ends, as well as forthe case of medium risk (curve 84), where resampled efficient frontiersare used for discrimination of need-to-trade, in accordance with thepresent invention. In each instance, a region of nearly symmetric‘tolerance’ exists where reasonably diversified portfolios aredetermined to be statistically similar to optimized targets.

Consequently, in accordance with preferred embodiments of the invention,an associated meta-resampled efficient frontier is found, as describedabove, for each classical efficient frontier corresponding to asimulation of returns. This procedure may advantageously lead to arebalancing test of greater uniformity, particularly at the high-riskend of the resampled efficient frontier, and thereby advantageouslyproviding for automation of some or all rebalancing decisions if sodesired.

Additionally, in certain circumstances, it is desirable to control thepower of a rebalancing test across the spectrum of portfolio risk inaccordance with particular investment objectives and strategies,objectives, and end-user requirements. One means to that end is ensuringthe statistical relevance of the ensemble of statistically weightedportfolios against which a CURRENT portfolio is tested. To that end, twoparameters are significant: the percentage of relevant simulations tokeep for purposes of statistical comparison with the CURRENT portfolio,and a ‘relevance’ criterion governing the number of portfolios retainedwithin the ensemble per simulation. The first parameter, generally, hasthe effect of increasing the power of a rebalancing test as fewersimulations are kept. With respect to the second parameter, increasingthe number of portfolios per simulation reduces the power of the testwhile, at the same time, typically spreading out the power over anincreased range of portfolio risk.

Restriction of the percentage of simulations, or of the consideredportfolios per simulation, for statistical consideration may be furtherrefined by averaging simulations to yield meta-resampled efficientfrontiers, thereby deriving benefits associated with comparing resampledportfolios with resampled, rather than classically optimized,portfolios, as described above.

Estimating Normal Ranges of Portfolio Weights

Defining an investment-relevant normal range of portfolio weightsprovides useful guidelines for many asset management functions includingat-a-glance identification of anomalous portfolio structure andinstances in which portfolio rebalancing to optimality may be advisable.

For each asset of a target optimal portfolio, an enhanced estimate ofthe normal range of asset weights can be calculated. The rangeincorporates the distribution of the asset weights of the associatedmeta-resampled or bootstrapped resampled portfolios. In a preferredembodiment, the associated meta-resampled portfolios asset weights foreach asset of the target portfolio form a meta-resampled distribution ofasset weights. Various descriptive statistical measures can then beapplied to provide an estimate of a normal range relative to each assetof the target optimal portfolio. For example, in the table below, the25th and 75th percentile values of the meta-resampled distribution ofasset weights for each asset for ten asset classes for medium and highrisk resampled efficient target portfolios based on historical returndata are given. Unlike current asset range estimate methods, resampledmethods provide investment relevant estimates that vary in statisticalcharacteristics by asset and target portfolio risk.

Resampled Index-Associated Portfolio Weights Range Estimates Medium RiskHigh Risk 25th 75th 25th 75th Pctile Pctile Pctile Pctile Money Market10%  22% 0%  0% Intermediate Fixed 12%  34% 0%  1% Long Term Fixed 3%14% 1% 12% High Yield Corp 3% 19% 0%  4% Large Cap Value 2% 13% 1% 17%Large Cap Growth 0%  5% 1% 12% Small/Mid Cap Value 0%  3% 0%  7%Small/Mid Cap 0%  4% 4% 35% Growth International Stocks 2% 14% 3% 30%Real Estate 3% 15% 2% 25%

In alternative embodiments, the disclosed methods for evaluating anexisting or putative portfolio may be implemented as a computer programproduct for use with a computer system. Such implementations may includea series of computer instructions fixed either on a tangible medium,such as a computer readable medium (e.g., a diskette, CD-ROM, ROM, orfixed disk) or transmittable to a computer system, via a modem or otherinterface device, such as a communications adapter connected to anetwork over a medium. The medium may be either a tangible medium (e.g.,optical or analog communications lines) or a medium implemented withwireless techniques (e.g., microwave, infrared or other transmissiontechniques). The series of computer instructions embodies all or part ofthe functionality previously described herein with respect to thesystem. Those skilled in the art should appreciate that such computerinstructions can be written in a number of programming languages for usewith many computer architectures or operating systems. Furthermore, suchinstructions may be stored in any memory device, such as semiconductor,magnetic, optical or other memory devices, and may be transmitted usingany communications technology, such as optical, infrared, microwave, orother transmission technologies. It is expected that such a computerprogram product may be distributed as a removable medium withaccompanying printed or electronic documentation (e.g., shrink wrappedsoftware), preloaded with a computer system (e.g., on system ROM orfixed disk), or distributed from a server or electronic bulletin boardover the network (e.g., the Internet or World Wide Web). Of course, someembodiments of the invention may be implemented as a combination of bothsoftware (e.g., a computer program product) and hardware. Still otherembodiments of the invention are implemented as entirely hardware, orentirely software (e.g., a computer program product).

Once an optimized set of portfolio weights is determined according to aspecified risk objective and in accordance with the foregoing teachings,funds are invested in accordance with the portfolio weights that havebeen determined.

The described embodiments of the invention are intended to be merelyexemplary and numerous variations and modifications will be apparent tothose skilled in the art. All such variations and modifications areintended to be within the scope of the present invention as defined inthe appended claims.

1. A computer-implemented method for selecting a value of a portfolioweight for each of a plurality of assets of an optimal portfolio, thevalue of portfolio weight chosen from specified values between realnumbers c₁ and c₂ associated with each asset, each asset characterizedby an expected return and a standard deviation of return, and acovariance with respect to each of every other asset of the plurality ofassets, the method comprising: a. choosing a forecast certainty levelfrom a collection of forecast certainty levels for defining a resamplingprocess of the input data consistent with the assumed forecast certaintyof input data characterizing the expected return and standard deviationof return of each of the plurality of assets; b. generating a pluralityof optimization inputs drawn at least from a distribution of simulatedoptimization inputs consistent with the expected return, the standarddeviation of return of each of the plurality of assets and the chosenforecast certainty level; c. computing a simulated mean-varianceefficient frontier, the frontier having at least one simulatedmean-variance efficient portfolio, for each of the plurality ofoptimization inputs; d. associating each mean-variance efficientportfolio with a specified set of simulated mean-variance efficientportfolios for creating a set of associated mean-variance efficientportfolios; e. establishing a statistical mean for each set ofassociated mean-variance efficient portfolios, thereby generating aplurality of statistical means, the plurality of statistical meansdefining a meta-resampled efficient frontier; f. selecting a portfolioweight for each asset from the meta-resampled efficient frontieraccording to a specified investment objective for defining the optimalportfolio subject to the specified investment objective.
 2. A methodaccording to claim 1, wherein the specified investment objective is arisk objective.
 3. A method according to claim 1, wherein the step ofchoosing a forecast certainty level from a collection of forecastcertainty levels includes choosing from a set of indices from 1 to N,where each index is calibrated to provide a different level of forecastcertainty.
 4. A method according to claim 3, further includingincreasing the number of samples drawn from a distribution based on ahigher level of forecast certainty.
 5. A method according to claim 3,wherein each level represent a geometric increase (or decrease) in thenumber of observations drawn from the distribution.
 6. A methodaccording to claim 1, wherein the step of associating each mean-varianceefficient portfolio with a specified set of simulated mean-varianceefficient portfolios includes associating on the basis of proximity to amaximized expected utility.
 7. A method according to claim 6, whereinthe expected utility is a quantity μ−λσ².
 8. A method according to claim1, wherein the step of associating each mean-variance efficientportfolio with a specified set of simulated mean-variance efficientportfolios includes associating on the basis of a measure of risk.
 9. Amethod according to claim 8, wherein the measure of risk is variance.10. A method according to claim 8, wherein the measure of risk isresidual risk.
 11. A method according to claim 1, wherein the step ofassociating each mean-variance efficient portfolio with a specified setof simulated mean-variance efficient portfolios includes associating onthe basis of a measure of return.
 12. A method according to claim 11,wherein the measure of return is expected return.
 13. A method accordingto claim 11, wherein the measure of return is expected residual return.14. A method according to claim 1, wherein the step of associating eachmean-variance efficient portfolio with a specified set of simulatedmean-variance efficient portfolios includes associating on a basis ofproximity of the portfolios to other simulated mean-variance efficientportfolios.
 15. A method according to claim 1, wherein the step ofassociating each mean-variance efficient portfolio with a specified setof simulated mean-variance efficient portfolios includes associating ona basis of ranking by a specified criterion, the criterion chosen from agroup including risk and expected return.
 16. A method according toclaim 1, wherein the step of associating each mean-variance efficientportfolio with a specified set of simulated mean-variance efficientportfolios includes associating on a basis of investment relevance ofthe simulated mean-variance efficient portfolios.
 17. A method accordingto claim 1, wherein the step of associating each mean-variance efficientportfolio with a specified set of simulated mean-variance efficientportfolios includes associating simulated portfolios based on aselection of relevant simulations.
 18. A method according to claim 1,wherein the step of associating each mean-variance efficient portfoliowith a specified set of simulated mean-variance efficient portfoliosincludes specifying a fraction of portfolios per simulation to beconsidered in the association.
 19. A method according to claim 1,further including calculating an estimated normal range of asset weightsfor an optimal portfolio.
 20. A method according to claim 1, furthercomprising calculating a portfolio rebalancing probability for a givenportfolio with respect to an optimal portfolio.
 21. A method accordingto claim 1, further comprising investing funds in accordance with theselected portfolio weights.
 22. A computer program product for use on acomputer system for selecting a value of portfolio weight for each of aspecified plurality of assets of an optimal portfolio, the value ofportfolio weight chosen from values between two specified real numbers,each asset having an expected return and a standard deviation of return,and each asset having a covariance with respect to each of every otherasset of the plurality of assets, the computer program productcomprising a computer usable medium having computer readable programcode thereon, the computer readable program code including: a. a routinefor generating a plurality of optimization inputs drawn from adistribution of simulated optimization inputs statistically consistentwith the expected return and the standard deviation of return of each ofthe plurality of assets; b. program code for computing a simulatedmean-variance efficient frontier, the frontier comprising at least onesimulated mean-variance efficient portfolio for each of the plurality ofoptimization inputs; c. program code for associating each simulatedmean-variance efficient portfolio with a specified set of portfolios forcreating a set of associated mean-variance efficient portfolios; d. amodule for establishing a statistical mean for each set of associatedmean-variance efficient portfolios, the plurality of statistical meansdefining a meta-resampled efficient frontier; and e. program code forselecting a portfolio weight for each asset from the meta-resampledefficient frontier according to a specified risk objective.
 23. Acomputer program product according to claim 22, wherein the routine forresampling a plurality of simulations on input data draws a number ofreturns from a simulation based on a specified level of estimationcertainty.
 24. A computer program product according to claim 22, whereinthe set of associated mean-variance efficient portfolios is based on aspecified level of forecast certainty.